The Eagles are desperate to sign a targetman in the January transfer window after the injury to Christian Benteke that looks set to keep him out until well into the New Year.
Tosun, 26, has been in terrific form for Besiktas – where he scored 20 league goals to fire them to the Turkish title last season.
Palace fell short with a bid for the German-born forward in the summer and he has been impressing this season in the Champions League.
However, that could prove to be a problem if Besiktas, who have six points from their opening two matches, qualify for the knockout phase of the competition.
They could delay any sale of Tosun until the end of the season, which would mean Roy Hodgson will have to look elsewhere for a striker.
Newcastle United and Tottenham Hotspur are also tracking Tosun and posted scouts to Eskisehir on Friday for the World cup qualifier, which Iceland won 3-0.
Borussia Dortmund are reported to have dropped all interest in Chelsea striker Michy Batshuayi and are looking elsewhere.
Sevilla are bracing themselves for interest in Steven Nzonzi and have lined up a replacement in the form of Marcos Llorente.
Newcastle are targeting a move for Juventus midfielder Stefan Sturaro and are reported to have lodged an opening bid of £11.5million.
Southampton have joined the race to sign Norwich midfielder James Maddison, but will have to move quickly to secure his signing.
Bournemouth are weighing up a move for Sheffield United midfielder David Brooks, but will have to pay £12million to land him.
Follow the final World Cup game of the weekend LIVE as Brazil face Switzerland in Rostov.